D₄×AGL(1,ℤ/25) #72228811

order	4000
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

D₄×AGL(1,ℤ/25)

Splittings

Fitting	D₄×5₋¹⁺² ⋊ C₄
Sylow	5₋¹⁺² ⋊ C₄×D₄

Subgroups

	subgroup	G/H
center	C₂	C₂²×AGL(1,ℤ/25)
commutator	C₂×C₂₅	C₂²×C₄×C₅
Frattini, socle	C₂×C₅	C₂²×C₅×AGL(1,𝔽₅)
Fitting	D₄×5₋¹⁺²	C₄
prime radical	D₄×D₂₅⋊C₅	C₂
cosocle	C₂×D₂₅	C₂³×C₅
hypercenter	D₄	AGL(1,ℤ/25)
hypocenter	C₂₅	C₄×C₅×D₄
LCS	G ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂ ◁ D₄

290 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 100

A minimal faithful permutation representation — 3 generators on 29 points:

(2,12,25,7,8,4,17,23,19,21,6,18,16,15,14,13,5,22,24,20)(3,9,10,11);(1,5,20,22,25,3,2,18,4,24,11,8,7,15,14,9,19,13,12,6,10,16,23,21,17)(27,29);(26,27,28,29)

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	S₄×AGL(1,ℤ/25), C₂×D₄×AGL(1,ℤ/25), #41×AGL(1,ℤ/25), …
maximal normal subgroup of	C₂×D₄×AGL(1,ℤ/25), #41×AGL(1,ℤ/25), #72275748, …
maximal quotient of	C₂×D₄×AGL(1,ℤ/25), C₂²⋊C₄×AGL(1,ℤ/25), #72275641, …
automorphism group of	Dic₅₀, D₁₀₀, C₂×Dic₂₅
holomorph of	C₄×C₂₅

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((2,12,25,7,8,4,17,23,19,21,6,18,16,15,14,13,5,22,24,20)(3,9,10,11), (1,5,20,22,25,3,2,18,4,24,11,8,7,15,14,9,19,13,12,6,10,16,23,21,17)(27,29), (26,27,28,29));